She again declined to place the blame for the debacle, which has caught up thousands of people and is widely regarded as a result of her own “hostile environment” strategy, on individual ministers or civil servants at her former department.

Sajid Javid, appointed home secretary on Monday morning following Rudd’s departure, vowed to make sure people affected by the Windrush debacle were treated with “decency and fairness” as he arrived at the Home Office to start his new job.

Javid said he would look carefully at the government’s immigration policy, although No 10 insisted there were no plans to change direction on the issue.

May, on a local elections campaign visit, said it was right that the government continued to deal with illegal immigration through enforced removals, a policy that also existed during her tenure at the Home Office.

“When I was home secretary, yes, there were targets in terms of removing people from the country, who were here illegally,” she told Sky News.

However, she failed to acknowledge that her hostile environment policy may have also had a severe impact on the Windrush generation, many of whom struggled to prove they were in the UK legally.

“Amber Rudd was very clear about the reasons why she has resigned – that was because of information she gave to the House of Commons which was not correct,” she said.

“If you look at what we’re doing as a government, and have been doing over the years as a government, what we are doing is responding to the need that people see for a government to deal with illegal immigration.

“That’s exactly what we are doing. Now, we have seen the Windrush generation being caught up in way that has caused anxiety among that generation. That’s why we have set up a unit that is helping those people top get the documents that they need.”

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“They are British, they are part of us. But we deal with that, we make sure that people are given the reassurance that they need, but we also need to ensure that we’re dealing with illegal immigration.”

Jeremy Corbyn said the chaos at the Home Office was “deeply unsatisfactory” and that Rudd’s departure made it even more pressing for May to be brought before parliament to answer questions on the affair.

“Amber Rudd has been the human shield for Theresa May, and she’s now gone,” he said. “Theresa May now has questions to answer – from the liaison committee or wherever else those questions are raised – about what she actually did as home secretary and what she said.”

With Downing Street seeking to protect May, her official spokesman stressed that she would not have been expected to be across the detail of Home Office targets once she had left the department.

“The prime minister was aware when she was in office as home secretary of targets, but what the prime minister would have ceased to receive when she moved over to Downing Street was that sort of operational detail,” he said.

Javid, the former communities secretary, said his “most urgent task” was to get to grips with the Windrush crisis, and ensure those affected “are all treated with the decency and the fairness they deserve”.

The new home secretary said he recognised the scandal could cause concern among ethnic minority voters but issued a plea to those who were wavering to look at the government’s attempts to “put things right”.

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Yet he declined to say whether this might bring an end to the hostile environment immigration policy or targets for net removals.

Rudd was forced to step down after a series of revelations in the Guardian about problems faced by Windrush-generation residents, which culminated in a leak on Friday that appeared to show she was aware of targets for removing people who were in the country illegally.

The pressure increased on Sunday afternoon as the Guardian revealed that in a leaked 2017 letter to May, Rudd had told the prime minister of her intention to increase deportations by 10% – seemingly at odds with her recent denials that she was aware of deportation targets.

Rudd, the fifth enforced departure from the cabinet since last year’s snap general election, admitted when she stood down that she had “inadvertently” misled MPs over government targets for removing illegal migrants. But she had also been battling intense criticism over the Windrush scandal.